Covariance of Maxwell Eqn. - conceptual question

This has been bugging me for quite a while now. My question is essentially about how one shows that Maxwell equations are invariant under Lorentz transforms. Writing them in index notation, it is usually appealed to that all terms involved are Lorentz tensors (or contractions thereof), and therefore the covariance is manifest. This does, however, assume that we know the transformation properties of the objects in the equation i.e that they are tensorial.

For example, writing the source equations as:

d_u F^(uv) = j^v

we do indeed know that d_u transforms as a covariant 4-vector, directly from the definition of LT. We can also construct an argument to should that j^v (which has as components the charge density and current density) is a contravariant 4-vector, by considering how densities change under LT (i.e. length contraction) etc. However, the question still remains that we need to first show that F^(uv) is a tensor in order to state that Maxwell's equations are indeed invariant under LT. This problem reduces to showing that the E and B fields transform in certain ways, but how does one derive these transformation properties? All I can find in books and on the internet is people deriving the E and B field transformation properties FROM the tensorial nature of F^(uv) itself, which is viciously circular if you are trying to show that Maxwell's equations.

I have found a few specific arguments of people seeming to find these E and B transformation properties for specific physical situations, but nothing too convincing. I looked at Einstein's paper on SR (not the original 1905 one, but one a few years later), and in it he in fact essentially says "Maxwell's equations are invariant under LT if we assume that E and B transform as the following...". I cannot see how, without some other verification (theoretical or experimental) for those transformation laws, one can state that Maxwell's equations are invariant under LT.

Any help with this would be much appreciated - it's been bugging me for so long!

P.S. Please let me know if I should post this in the "Electrodynamics" section instead!

Would it suffice to you if we start with some hypothetical equation [itex]\partial_\mu F^{\mu \nu} = j^\nu[/itex] and show that this equation yields the Maxwell equations, therefore the transformations of the components of [itex]F[/itex] directly tell us about the transformations of the electric and magnetic fields?

Thanks for your message. I cannot see how that helps, because identifying that equation with Maxwell's equations is fine, but that is all within a single reference frame. As soon as you go to another frame, you need to know how F transforms to be able to determine how E and B transform (and as far as I can tell, you need to know how E and B transform to actually be able to know how F transforms, i.e. tensorially, which is circular).

Thanks for your message. I cannot see how that helps, because identifying that equation with Maxwell's equations is fine, but that is all within a single reference frame. As soon as you go to another frame, you need to know how F transforms to be able to determine how E and B transform (and as far as I can tell, you need to know how E and B transform to actually be able to know how F transforms, i.e. tensorially, which is circular).

This is by no means unique to you, but I think people in general get hung up on transforming between reference frames a bit too much. It's the SR equivalent of a rotation, and how often have you second-guessed how a vector transforms under a rotation?

(Admittedly, it's not wrong to wonder given that the entire topic is about electric and magnetic field vectors transforming, but my point is that this is like asking how the x and y components of a vector transform under a rotation--you get complicated formulas that mix the two together, but when you consider the vector as a whole, the picture becomes a lot more coherent.)

At any rate, I think the clearest picture is to start with some rank-2 tensor [itex]F[/itex] that obeys [itex]\partial_u F^{\mu \nu} = j^\nu[/itex], identify that the various equations with respect to certain components generate Maxwell's equations, and from there conclude that F, which you already posited was a tensor, is the the unique object that combines the electric and magnetic fields in a covariant manner.

Basically, you start with how F transforms and conclude how E and B transform as a result. This is not as sketchy as it sounds because all true tensors of the same signature transform in the same way; the bigger leap is puzzling out what kind of tensor subsumes the electric and magnetic fields.